Differentiability and Differentiation

The rate of change of a quantity y with respect to another quantity x is called the derivative of y with respect to x.

The derivative of f, denoted by f'(x) is given by f'(x) = limΔx→0 (Δy)/(Δ x) = dy/dx

The right hand derivative of f at x = a is denoted by f'(a+) and is given by f'(a+) = limh→0+(f(a+h)-f(a))/h

The left hand derivative of f at x = a is denoted by f'(a-) and is given by f'(a-) = limh→0- (f(a-h)-f(a))/-h

For a function to be differentiable at x=a, we should have f'(a-) = f'(a+) i.e. limh→0 (f (a-h)-f (a))/ (-h) = limh→0 (f(a+h) – f (a))/h.

limh→0 sin 1/h fluctuates between -1 and 1.

If at a particular point say x =a, we have f'(a+) = t1 (a finite number) and f'(a-) = t2 (a finite number) and if t1 ≠ t2, then f' (a) does not exist, but f(x) is a continuous function at x = a.

Continuity and differentiability are quite interrelated. Differentiabilityalways implies continuity but the converse is not true. This means that a differentiable function is always continuous but if a function is continuous it may or may not be differentiable.

Some basic formulae:

d/dx (logex) = 1/x, x > 0

d/dx (logax) = 1/x logea

d/dx (sin x) = cos x

d/dx (cos x) = – sin x

d/dx (tan x) = sec2x, x ≠ (2n+1) π/2, n∈ I.

d/dx (cot x) = - cosec2x, x ≠ nπ, n∈ I.

d/dx (sec x) = sec x tan x, x ≠ (2n+1) π/2, n∈ I.

d/dx (cosec x) = - cosec x cot x, x ≠ nπ, n∈ I.

d/dx (sin-1 x) = 1/√(1 – x2), -1 < x < 1

d/dx (cos-1 x) = – 1/√(1 – x2), -1 < x < 1

d/dx (tan-1 x) = 1/(1 + x2)

d/dx (cot-1 x) = - 1/(1 + x2)

d/dx (cosec-1 x) = - 1/|x|√(x2 – 1), |x| > 1

d/dx (sec-1 x) = 1/|x|√(x2 – 1), |x| > 1

d/dx (sinh x) = cosh x

d/dx (cosh x) = sinh x

d/dx (tanh x) = sech2x

d/dx (coth x) = - cosech2x

d/dx (sech x) = – sech x tanh x

d/dx (cosech x) = - cosech x coth x

If a function is not derivable at a point, it need not imply that it is discontinuous at that point. But, however, discontinuity at a point necessarily implies non-derivability.

In case, a function is not differentiable but is continuous at a particular point say x = a, then it geometrically implies a sharp corner at x = a.

A function f is said to be derivable over a closed interval [a, b] if :

For the points a and b, f'(a+) and f'(b-) exist and

For ant point c such that a < c < b, f'(c+) and f'(c-) exist and are equal.

If y = f(u) and u = g(x), then dy/dx = dy/du.du/dx = f'(g(x)) g'(x). This method is also termed as the chain rule.

For composite functions, differentiation is carried out in this way:

If y = [f(x)]n, then we put u = f(x). So that y = un. Then by chain rule:

dy/dx = dy/du.du/dx = nu(n-1)f' (x) = [f(x)](n-1) f' (x)

If the functions f(x) and g(x) are derivable at x = a, then the following functions are also derivable:

f(x) + g(x)

f(x) - g(x)

f(x) . g(x)

f(x) / g(x), provided g(a) ≠ 0

If the function f(x) is differentiable at x = a while g(x) is not derivable at x = a, then the product function f(x). g(x) can still be differentiable at x = a.

Even if both the functions f(x) and g(x) are not differentiable at x = a, the product function f(x).g(x) can still be differentiable at x = a.

Even if both the functions f(x) and g(x) are not derivable at x = a, the sum function f(x) + g(x) can still be differentiable at x = a.

If function f(x) is derivable at x = a, this need not imply that f'(x) is continuous at x = a.

Differentiation using substitution: The following substitutions may be used for computing the differentiation of the functions:

√a2 - x2, use x = a sin θ or a cos θ

√a2 + x2, use x = a tan θ or a cot θ

√x2 - a2, use x = a sec θ or a cosec θ

√(a + x) or √(a – x) , use x = a cos 2θ

√2ax- x2, use x = a (1 – cos θ)

If x = f(t) and y = g(t), where t is a parameter, then dy/dx = (dy/dt)/(dx/dt) = g’(t)/f’(t)

If a function is in the form of exponent of a function over another function such as f(x)g(x), we first take logarithm and then differentiate.